On the Rank of Z <inf>8</inf> -linear Hadamard Codes

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Abstract

© 2018 Elsevier B.V. The Z 2 s -additive codes are subgroups of Z 2 sn , and can be seen as a generalization of linear codes over Z 2 and Z 4 . A Z 2 s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z 2 s -additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z 4 -linear Hadamard codes. However, when s>2, the dimension of the kernel of Z 2 s -linear Hadamard codes of length 2 t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is given for s=3. Moreover, it is shown that this invariant, along with the dimension of the kernel, provides a complete classification, once t≥3 is fixed. In this case, the number of nonequivalent such codes is also established.
Original languageEnglish
Pages (from-to)25-30
JournalElectronic Notes in Discrete Mathematics
Volume70
DOIs
Publication statusPublished - 1 Dec 2018

Keywords

  • classification
  • Gray map
  • Hadamard code
  • Kernel
  • Rank
  • Z -additive code 2 s

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